Fitting SSF models to the buffalo data
Dynamic step selection functions with temporal harmonics - wet season
Load packages
Importing buffalo data
Import the buffalo data with random steps and extracted covariates that we created for the paper Forrest et al. (2024), in the script Ecography_DynamicSSF_1_Step_generation. This repo can be found at: swforrest/dynamic_SSF_sims.
Here we create the sine and cosine terms that were interact with each of the covariates to fit temporally varying parameters.
Importing data
Rows: 1165406 Columns: 22
── Column specification ────────────────────────────────────────────────────────
Delimiter: ","
dbl (18): id, burst_, x1_, x2_, y1_, y2_, sl_, ta_, dt_, hour_t2, step_id_,...
lgl (1): case_
dttm (3): t1_, t2_, t2_rounded
ℹ Use `spec()` to retrieve the full column specification for this data.
ℹ Specify the column types or set `show_col_types = FALSE` to quiet this message.Importing data
buffalo_data_all <- buffalo_data_all %>%
mutate(t1_ = lubridate::with_tz(buffalo_data_all$t1_, tzone = "Australia/Darwin"),
t2_ = lubridate::with_tz(buffalo_data_all$t2_, tzone = "Australia/Darwin"))
buffalo_data_all <- buffalo_data_all %>%
mutate(id_num = as.numeric(factor(id)),
step_id = step_id_,
x1 = x1_, x2 = x2_,
y1 = y1_, y2 = y2_,
t1 = t1_,
t1_rounded = round_date(buffalo_data_all$t1_, "hour"),
hour_t1 = hour(t1_rounded),
t2 = t2_,
t2_rounded = round_date(buffalo_data_all$t2_, "hour"),
hour_t2 = hour(t2_rounded),
hour = hour_t2,
yday = yday(t1_),
year = year(t1_),
month = month(t1_),
sl = sl_,
log_sl = log(sl_),
ta = ta_,
cos_ta = cos(ta_),
# scale canopy cover from 0 to 1
canopy_01 = canopy_cover/100,
# here we create the harmonic terms for the hour of the day
# for seasonal effects, change hour to yday (which is tau in the manuscript),
# and 24 to 365 (which is T)
hour_s1 = sin(2*pi*hour/24),
hour_s2 = sin(4*pi*hour/24),
hour_s3 = sin(6*pi*hour/24),
hour_c1 = cos(2*pi*hour/24),
hour_c2 = cos(4*pi*hour/24),
hour_c3 = cos(6*pi*hour/24))
# to select a single year of data
# buffalo_data_all <- buffalo_data_all %>% filter(t1 < "2019-07-25 09:32:42 ACST")
buffalo_ids <- unique(buffalo_data_all$id)
# Timeline of buffalo data
buffalo_data_all %>% ggplot(aes(x = t1, y = factor(id), colour = factor(id))) +
geom_point(alpha = 0.1) +
scale_y_discrete("Buffalo ID") +
scale_x_datetime("Date") +
scale_colour_viridis_d() +
theme_bw() +
theme(legend.position = "none")
Fitting the models
Creating a data matrix
First we create a data matrix to be provided to the model, and then we scale and centre the full data matrix, with respect to each of the columns. That means that all variables are scaled and centred after the data has been split into wet and dry season data, and also after creating the quadratic and harmonic terms (when using them).
We should only include covariates in the data matrix that will be used in the model formula.
Models
- 0p = 0 pairs of harmonics
- 1p = 1 pair of harmonics
- 2p = 2 pairs of harmonics
- 3p = 3 pairs of harmonics
For the dynamic models, we start to add the harmonic terms. As we have already created the harmonic terms for the hour of the day (s1, c1, s2, etc), we just interact (multiply) these with each of the covariates, including the quadratic terms, prior to model fitting. We store the scaling and centering variables to reconstruct the natural scale coefficients.
To provide some intuition about harmonic regression we have created a walkthrough script for Forrest et al. (2024), in the script Ecography_DynamicSSF_Walkthrough_Harmonics_and_selection_surfaces, which can be found at: swforrest/dynamic_SSF_sims, that introduces harmonics and how they can be used to model temporal variation in the data. It will help provide some understand the model outputs and how we construct the temporally varying coefficients in this script.
Selecting data
Code
months_wet <- c(1:4, 11:12)
buffalo_ids <- unique(buffalo_data_all$id)
focal_id <- 2005
# comment and uncomment the relevant lines to get either wet or dry season data
buffalo_data <- buffalo_data_all %>% filter(id == focal_id & month %in% months_wet) # wet season
# buffalo_data <- buffalo_data_all %>% filter(id == focal_id & !month %in% months_wet) # dry season
# all data
# buffalo_data <- buffalo_data_all %>% filter(id == focal_id)Code
buffalo_data_matrix_unscaled <- buffalo_data %>% transmute(
ndvi = ndvi_temporal,
ndvi_sq = ndvi_temporal ^ 2,
canopy = canopy_01,
canopy_sq = canopy_01 ^ 2,
slope = slope,
herby = veg_herby,
step_l = sl,
log_step_l = log_sl,
cos_turn_a = cos_ta)
buffalo_data_matrix_scaled <- scale(buffalo_data_matrix_unscaled)
# save the scaling values to recover the natural scale of the coefficients
# which is required for the simulations
# (so then environmental variables don't need to be scaled)
mean_vals <- attr(buffalo_data_matrix_scaled, "scaled:center")
sd_vals <- attr(buffalo_data_matrix_scaled, "scaled:scale")
scaling_attributes_0p <- data.frame(variable = names(buffalo_data_matrix_unscaled),
mean = mean_vals, sd = sd_vals)
# add the id, step_id columns and presence/absence columns to
# the scaled data matrix for model fitting
buffalo_data_scaled_0p <- data.frame(id = buffalo_data$id,
step_id = buffalo_data$step_id,
y = buffalo_data$y,
buffalo_data_matrix_scaled)Code
buffalo_data_matrix_unscaled <- buffalo_data %>% transmute(
# the 'linear' term
ndvi = ndvi_temporal,
# interact with the harmonic terms
ndvi_s1 = ndvi_temporal * hour_s1,
ndvi_c1 = ndvi_temporal * hour_c1,
ndvi_sq = ndvi_temporal ^ 2,
ndvi_sq_s1 = (ndvi_temporal ^ 2) * hour_s1,
ndvi_sq_c1 = (ndvi_temporal ^ 2) * hour_c1,
canopy = canopy_01,
canopy_s1 = canopy_01 * hour_s1,
canopy_c1 = canopy_01 * hour_c1,
canopy_sq = canopy_01 ^ 2,
canopy_sq_s1 = (canopy_01 ^ 2) * hour_s1,
canopy_sq_c1 = (canopy_01 ^ 2) * hour_c1,
slope = slope,
slope_s1 = slope * hour_s1,
slope_c1 = slope * hour_c1,
herby = veg_herby,
herby_s1 = veg_herby * hour_s1,
herby_c1 = veg_herby * hour_c1,
step_l = sl,
step_l_s1 = sl * hour_s1,
step_l_c1 = sl * hour_c1,
log_step_l = log_sl,
log_step_l_s1 = log_sl * hour_s1,
log_step_l_c1 = log_sl * hour_c1,
cos_turn_a = cos_ta,
cos_turn_a_s1 = cos_ta * hour_s1,
cos_turn_a_c1 = cos_ta * hour_c1)
buffalo_data_matrix_scaled <- scale(buffalo_data_matrix_unscaled)
mean_vals <- attr(buffalo_data_matrix_scaled, "scaled:center")
sd_vals <- attr(buffalo_data_matrix_scaled, "scaled:scale")
scaling_attributes_1p <- data.frame(variable = names(buffalo_data_matrix_unscaled),
mean = mean_vals, sd = sd_vals)
buffalo_data_scaled_1p <- data.frame(id = buffalo_data$id,
step_id = buffalo_data$step_id,
y = buffalo_data$y,
buffalo_data_matrix_scaled)Code
buffalo_data_matrix_unscaled <- buffalo_data %>% transmute(
ndvi = ndvi_temporal,
ndvi_s1 = ndvi_temporal * hour_s1,
ndvi_s2 = ndvi_temporal * hour_s2,
ndvi_c1 = ndvi_temporal * hour_c1,
ndvi_c2 = ndvi_temporal * hour_c2,
ndvi_sq = ndvi_temporal ^ 2,
ndvi_sq_s1 = (ndvi_temporal ^ 2) * hour_s1,
ndvi_sq_s2 = (ndvi_temporal ^ 2) * hour_s2,
ndvi_sq_c1 = (ndvi_temporal ^ 2) * hour_c1,
ndvi_sq_c2 = (ndvi_temporal ^ 2) * hour_c2,
canopy = canopy_01,
canopy_s1 = canopy_01 * hour_s1,
canopy_s2 = canopy_01 * hour_s2,
canopy_c1 = canopy_01 * hour_c1,
canopy_c2 = canopy_01 * hour_c2,
canopy_sq = canopy_01 ^ 2,
canopy_sq_s1 = (canopy_01 ^ 2) * hour_s1,
canopy_sq_s2 = (canopy_01 ^ 2) * hour_s2,
canopy_sq_c1 = (canopy_01 ^ 2) * hour_c1,
canopy_sq_c2 = (canopy_01 ^ 2) * hour_c2,
slope = slope,
slope_s1 = slope * hour_s1,
slope_s2 = slope * hour_s2,
slope_c1 = slope * hour_c1,
slope_c2 = slope * hour_c2,
herby = veg_herby,
herby_s1 = veg_herby * hour_s1,
herby_s2 = veg_herby * hour_s2,
herby_c1 = veg_herby * hour_c1,
herby_c2 = veg_herby * hour_c2,
step_l = sl,
step_l_s1 = sl * hour_s1,
step_l_s2 = sl * hour_s2,
step_l_c1 = sl * hour_c1,
step_l_c2 = sl * hour_c2,
log_step_l = log_sl,
log_step_l_s1 = log_sl * hour_s1,
log_step_l_s2 = log_sl * hour_s2,
log_step_l_c1 = log_sl * hour_c1,
log_step_l_c2 = log_sl * hour_c2,
cos_turn_a = cos_ta,
cos_turn_a_s1 = cos_ta * hour_s1,
cos_turn_a_s2 = cos_ta * hour_s2,
cos_turn_a_c1 = cos_ta * hour_c1,
cos_turn_a_c2 = cos_ta * hour_c2)
buffalo_data_matrix_scaled <- scale(buffalo_data_matrix_unscaled)
mean_vals <- attr(buffalo_data_matrix_scaled, "scaled:center")
sd_vals <- attr(buffalo_data_matrix_scaled, "scaled:scale")
scaling_attributes_2p <- data.frame(variable = names(buffalo_data_matrix_unscaled),
mean = mean_vals, sd = sd_vals)
buffalo_data_scaled_2p <- data.frame(id = buffalo_data$id,
step_id = buffalo_data$step_id,
y = buffalo_data$y,
buffalo_data_matrix_scaled)Code
buffalo_data_matrix_unscaled <- buffalo_data %>% transmute(
ndvi = ndvi_temporal,
ndvi_s1 = ndvi_temporal * hour_s1,
ndvi_s2 = ndvi_temporal * hour_s2,
ndvi_s3 = ndvi_temporal * hour_s3,
ndvi_c1 = ndvi_temporal * hour_c1,
ndvi_c2 = ndvi_temporal * hour_c2,
ndvi_c3 = ndvi_temporal * hour_c3,
ndvi_sq = ndvi_temporal ^ 2,
ndvi_sq_s1 = (ndvi_temporal ^ 2) * hour_s1,
ndvi_sq_s2 = (ndvi_temporal ^ 2) * hour_s2,
ndvi_sq_s3 = (ndvi_temporal ^ 2) * hour_s3,
ndvi_sq_c1 = (ndvi_temporal ^ 2) * hour_c1,
ndvi_sq_c2 = (ndvi_temporal ^ 2) * hour_c2,
ndvi_sq_c3 = (ndvi_temporal ^ 2) * hour_c3,
canopy = canopy_01,
canopy_s1 = canopy_01 * hour_s1,
canopy_s2 = canopy_01 * hour_s2,
canopy_s3 = canopy_01 * hour_s3,
canopy_c1 = canopy_01 * hour_c1,
canopy_c2 = canopy_01 * hour_c2,
canopy_c3 = canopy_01 * hour_c3,
canopy_sq = canopy_01 ^ 2,
canopy_sq_s1 = (canopy_01 ^ 2) * hour_s1,
canopy_sq_s2 = (canopy_01 ^ 2) * hour_s2,
canopy_sq_s3 = (canopy_01 ^ 2) * hour_s3,
canopy_sq_c1 = (canopy_01 ^ 2) * hour_c1,
canopy_sq_c2 = (canopy_01 ^ 2) * hour_c2,
canopy_sq_c3 = (canopy_01 ^ 2) * hour_c3,
slope = slope,
slope_s1 = slope * hour_s1,
slope_s2 = slope * hour_s2,
slope_s3 = slope * hour_s3,
slope_c1 = slope * hour_c1,
slope_c2 = slope * hour_c2,
slope_c3 = slope * hour_c3,
herby = veg_herby,
herby_s1 = veg_herby * hour_s1,
herby_s2 = veg_herby * hour_s2,
herby_s3 = veg_herby * hour_s3,
herby_c1 = veg_herby * hour_c1,
herby_c2 = veg_herby * hour_c2,
herby_c3 = veg_herby * hour_c3,
step_l = sl,
step_l_s1 = sl * hour_s1,
step_l_s2 = sl * hour_s2,
step_l_s3 = sl * hour_s3,
step_l_c1 = sl * hour_c1,
step_l_c2 = sl * hour_c2,
step_l_c3 = sl * hour_c3,
log_step_l = log_sl,
log_step_l_s1 = log_sl * hour_s1,
log_step_l_s2 = log_sl * hour_s2,
log_step_l_s3 = log_sl * hour_s3,
log_step_l_c1 = log_sl * hour_c1,
log_step_l_c2 = log_sl * hour_c2,
log_step_l_c3 = log_sl * hour_c3,
cos_turn_a = cos_ta,
cos_turn_a_s1 = cos_ta * hour_s1,
cos_turn_a_s2 = cos_ta * hour_s2,
cos_turn_a_s3 = cos_ta * hour_s3,
cos_turn_a_c1 = cos_ta * hour_c1,
cos_turn_a_c2 = cos_ta * hour_c2,
cos_turn_a_c3 = cos_ta * hour_c3)
buffalo_data_matrix_scaled <- scale(buffalo_data_matrix_unscaled)
mean_vals <- attr(buffalo_data_matrix_scaled, "scaled:center")
sd_vals <- attr(buffalo_data_matrix_scaled, "scaled:scale")
scaling_attributes_3p <- data.frame(variable = names(buffalo_data_matrix_unscaled),
mean = mean_vals, sd = sd_vals)
buffalo_data_scaled_3p <- data.frame(id = buffalo_data$id,
step_id = buffalo_data$step_id,
y = buffalo_data$y,
buffalo_data_matrix_scaled)Model formula
As we have already precomputed and scaled the covariates, quadratic terms and interactions with the harmonics, we just include each parameter as a linear predictor.
Code
formula_1p <- y ~
ndvi +
ndvi_s1 +
ndvi_c1 +
ndvi_sq +
ndvi_sq_s1 +
ndvi_sq_c1 +
canopy +
canopy_s1 +
canopy_c1 +
canopy_sq +
canopy_sq_s1 +
canopy_sq_c1 +
slope +
slope_s1 +
slope_c1 +
herby +
herby_s1 +
herby_c1 +
step_l +
step_l_s1 +
step_l_c1 +
log_step_l +
log_step_l_s1 +
log_step_l_c1 +
cos_turn_a +
cos_turn_a_s1 +
cos_turn_a_c1 +
strata(step_id) Code
formula_2p <- y ~
ndvi +
ndvi_s1 +
ndvi_s2 +
ndvi_c1 +
ndvi_c2 +
ndvi_sq +
ndvi_sq_s1 +
ndvi_sq_s2 +
ndvi_sq_c1 +
ndvi_sq_c2 +
canopy +
canopy_s1 +
canopy_s2 +
canopy_c1 +
canopy_c2 +
canopy_sq +
canopy_sq_s1 +
canopy_sq_s2 +
canopy_sq_c1 +
canopy_sq_c2 +
slope +
slope_s1 +
slope_s2 +
slope_c1 +
slope_c2 +
herby +
herby_s1 +
herby_s2 +
herby_c1 +
herby_c2 +
step_l +
step_l_s1 +
step_l_s2 +
step_l_c1 +
step_l_c2 +
log_step_l +
log_step_l_s1 +
log_step_l_s2 +
log_step_l_c1 +
log_step_l_c2 +
cos_turn_a +
cos_turn_a_s1 +
cos_turn_a_s2 +
cos_turn_a_c1 +
cos_turn_a_c2 +
strata(step_id) Code
formula_3p <- y ~
ndvi +
ndvi_s1 +
ndvi_s2 +
ndvi_s3 +
ndvi_c1 +
ndvi_c2 +
ndvi_c3 +
ndvi_sq +
ndvi_sq_s1 +
ndvi_sq_s2 +
ndvi_sq_s3 +
ndvi_sq_c1 +
ndvi_sq_c2 +
ndvi_sq_c3 +
canopy +
canopy_s1 +
canopy_s2 +
canopy_s3 +
canopy_c1 +
canopy_c2 +
canopy_c3 +
canopy_sq +
canopy_sq_s1 +
canopy_sq_s2 +
canopy_sq_s3 +
canopy_sq_c1 +
canopy_sq_c2 +
canopy_sq_c3 +
slope +
slope_s1 +
slope_s2 +
slope_s3 +
slope_c1 +
slope_c2 +
slope_c3 +
herby +
herby_s1 +
herby_s2 +
herby_s3 +
herby_c1 +
herby_c2 +
herby_c3 +
step_l +
step_l_s1 +
step_l_s2 +
step_l_s3 +
step_l_c1 +
step_l_c2 +
step_l_c3 +
log_step_l +
log_step_l_s1 +
log_step_l_s2 +
log_step_l_s3 +
log_step_l_c1 +
log_step_l_c2 +
log_step_l_c3 +
cos_turn_a +
cos_turn_a_s1 +
cos_turn_a_s2 +
cos_turn_a_s3 +
cos_turn_a_c1 +
cos_turn_a_c2 +
cos_turn_a_c3 +
strata(step_id)Fit the model
As we have already fitted the model, we will load it here, but if the model_fit file doesn’t exist, it will run the model fitting code. Be careful here that if you change the model formula, you will need to delete or rename the model_fit file to re-run the model fitting code, otherwise it will just load the previous model.
We are fitting a single model to the focal individual.
Code
if(file.exists(paste0("ssf_coefficients/model_id", focal_id, "_0p_harms_wet.rds"))) {
model_0p_harms <- readRDS(paste0("ssf_coefficients/model_id", focal_id, "_0p_harms_wet.rds"))
print("Read existing model")
} else {
tic()
model_0p_harms <- fit_clogit(formula = formula_0p,
data = buffalo_data_scaled_0p)
toc()
# save model object
saveRDS(model_0p_harms, file = paste0("ssf_coefficients/model_id", focal_id, "_0p_harms_wet.rds"))
print("Fitted model")
beep(sound = 2)
}[1] "Read existing model"$model
Call:
survival::clogit(formula, data = data, ...)
coef exp(coef) se(coef) z p
ndvi 0.31664 1.37250 0.07060 4.485 0.00000729038630065
ndvi_sq -0.26853 0.76450 0.07620 -3.524 0.000425
canopy -0.65865 0.51755 0.08418 -7.824 0.00000000000000512
canopy_sq 0.54151 1.71859 0.08199 6.604 0.00000000003989057
slope 0.01172 1.01179 0.02832 0.414 0.678946
herby 0.04652 1.04762 0.02628 1.771 0.076636
step_l -0.28465 0.75227 0.03014 -9.444 < 0.0000000000000002
log_step_l 0.29081 1.33752 0.02866 10.147 < 0.0000000000000002
cos_turn_a 0.02667 1.02703 0.01814 1.470 0.141504
Likelihood ratio test=229.7 on 9 df, p=< 0.00000000000000022
n= 39660, number of events= 3390
(864 observations deleted due to missingness)
$sl_
NULL
$ta_
NULL
$more
NULL
attr(,"class")
[1] "fit_clogit" "list" Code
if(file.exists(paste0("ssf_coefficients/model_id", focal_id, "_1p_harms_wet.rds"))) {
model_1p_harms <- readRDS(paste0("ssf_coefficients/model_id", focal_id, "_1p_harms_wet.rds"))
print("Read existing model")
} else {
tic()
model_1p_harms <- fit_clogit(formula = formula_1p,
data = buffalo_data_scaled_1p)
toc()
# save model object
saveRDS(model_1p_harms, file = paste0("ssf_coefficients/model_id", focal_id, "_1p_harms_wet.rds"))
print("Fitted model")
beep(sound = 2)
}[1] "Read existing model"$model
Call:
survival::clogit(formula, data = data, ...)
coef exp(coef) se(coef) z p
ndvi 0.27221 1.31287 0.09310 2.924 0.003456
ndvi_s1 -1.47785 0.22813 0.29048 -5.088 0.000000362757111271
ndvi_c1 -0.69794 0.49761 0.24641 -2.832 0.004620
ndvi_sq -0.27254 0.76144 0.09374 -2.907 0.003646
ndvi_sq_s1 0.70234 2.01847 0.18185 3.862 0.000112
ndvi_sq_c1 0.35512 1.42635 0.15957 2.226 0.026046
canopy -0.62154 0.53712 0.08764 -7.092 0.000000000001325872
canopy_s1 0.24421 1.27661 0.22771 1.072 0.283523
canopy_c1 -0.10677 0.89873 0.23101 -0.462 0.643951
canopy_sq 0.51114 1.66720 0.08511 6.006 0.000000001906791068
canopy_sq_s1 0.11711 1.12425 0.15777 0.742 0.457899
canopy_sq_c1 0.07700 1.08004 0.15768 0.488 0.625321
slope 0.01484 1.01495 0.02905 0.511 0.609436
slope_s1 -0.04321 0.95771 0.03778 -1.144 0.252717
slope_c1 -0.03944 0.96133 0.03931 -1.003 0.315737
herby 0.04505 1.04608 0.02745 1.641 0.100809
herby_s1 -0.01240 0.98768 0.05372 -0.231 0.817532
herby_c1 0.06057 1.06244 0.05728 1.057 0.290318
step_l -0.31739 0.72805 0.03131 -10.138 < 0.0000000000000002
step_l_s1 0.03119 1.03168 0.03702 0.842 0.399536
step_l_c1 0.04169 1.04257 0.03735 1.116 0.264398
log_step_l 0.35845 1.43111 0.03111 11.520 < 0.0000000000000002
log_step_l_s1 -0.47356 0.62278 0.05880 -8.054 0.000000000000000804
log_step_l_c1 -0.18578 0.83045 0.05681 -3.270 0.001075
cos_turn_a 0.03057 1.03104 0.01844 1.657 0.097449
cos_turn_a_s1 -0.05759 0.94404 0.01843 -3.125 0.001779
cos_turn_a_c1 0.02252 1.02278 0.01867 1.206 0.227667
Likelihood ratio test=434 on 27 df, p=< 0.00000000000000022
n= 39660, number of events= 3390
(864 observations deleted due to missingness)
$sl_
NULL
$ta_
NULL
$more
NULL
attr(,"class")
[1] "fit_clogit" "list" Code
if(file.exists(paste0("ssf_coefficients/model_id", focal_id, "_2p_harms_wet.rds"))) {
model_2p_harms <- readRDS(paste0("ssf_coefficients/model_id", focal_id, "_2p_harms_wet.rds"))
print("Read existing model")
} else {
tic()
model_2p_harms <- fit_clogit(formula = formula_2p,
data = buffalo_data_scaled_2p)
toc()
# save model object
saveRDS(model_2p_harms, file = paste0("ssf_coefficients/model_id", focal_id, "_2p_harms_wet.rds"))
print("Fitted model")
beep(sound = 2)
}[1] "Read existing model"$model
Call:
survival::clogit(formula, data = data, ...)
coef exp(coef) se(coef) z p
ndvi 0.316097 1.371763 0.097318 3.248 0.001162
ndvi_s1 -1.420877 0.241502 0.301480 -4.713 0.0000024408584
ndvi_s2 0.205236 1.227815 0.273278 0.751 0.452643
ndvi_c1 -0.632995 0.530999 0.306605 -2.065 0.038967
ndvi_c2 0.170273 1.185629 0.285123 0.597 0.550379
ndvi_sq -0.330068 0.718875 0.097580 -3.383 0.000718
ndvi_sq_s1 0.755142 2.127914 0.187450 4.029 0.0000561322922
ndvi_sq_s2 0.033228 1.033786 0.174182 0.191 0.848711
ndvi_sq_c1 0.292456 1.339713 0.190317 1.537 0.124372
ndvi_sq_c2 0.021747 1.021986 0.180771 0.120 0.904243
canopy -0.579578 0.560135 0.089387 -6.484 0.0000000000894
canopy_s1 0.382994 1.466669 0.235389 1.627 0.103722
canopy_s2 -0.149778 0.860899 0.231294 -0.648 0.517266
canopy_c1 0.008450 1.008486 0.237985 0.036 0.971677
canopy_c2 0.158279 1.171493 0.235046 0.673 0.500696
canopy_sq 0.471813 1.602897 0.086603 5.448 0.0000000509435
canopy_sq_s1 0.046310 1.047399 0.162272 0.285 0.775348
canopy_sq_s2 0.113950 1.120697 0.159173 0.716 0.474058
canopy_sq_c1 0.011185 1.011248 0.161156 0.069 0.944666
canopy_sq_c2 0.005665 1.005681 0.160571 0.035 0.971858
slope 0.012820 1.012902 0.031345 0.409 0.682554
slope_s1 -0.047972 0.953160 0.040111 -1.196 0.231697
slope_s2 -0.122505 0.884702 0.041155 -2.977 0.002914
slope_c1 -0.038410 0.962318 0.042722 -0.899 0.368612
slope_c2 0.029798 1.030246 0.040788 0.731 0.465050
herby 0.031493 1.031994 0.028253 1.115 0.264983
herby_s1 -0.026990 0.973371 0.055660 -0.485 0.627739
herby_s2 -0.029304 0.971121 0.055920 -0.524 0.600253
herby_c1 0.031413 1.031912 0.059584 0.527 0.598045
herby_c2 -0.156702 0.854959 0.056753 -2.761 0.005761
step_l -0.440720 0.643573 0.036136 -12.196 < 0.0000000000000002
step_l_s1 0.061805 1.063755 0.034710 1.781 0.074975
step_l_s2 -0.191891 0.825397 0.038767 -4.950 0.0000007428958
step_l_c1 -0.013252 0.986835 0.045126 -0.294 0.769004
step_l_c2 -0.212752 0.808357 0.040818 -5.212 0.0000001865672
log_step_l 0.443509 1.558165 0.033499 13.239 < 0.0000000000000002
log_step_l_s1 -0.636164 0.529319 0.067027 -9.491 < 0.0000000000000002
log_step_l_s2 -0.067998 0.934262 0.060114 -1.131 0.257991
log_step_l_c1 -0.107935 0.897686 0.059017 -1.829 0.067419
log_step_l_c2 -0.257057 0.773324 0.059509 -4.320 0.0000156299031
cos_turn_a 0.034114 1.034703 0.018741 1.820 0.068715
cos_turn_a_s1 -0.063351 0.938614 0.018750 -3.379 0.000728
cos_turn_a_s2 -0.054665 0.946802 0.018738 -2.917 0.003530
cos_turn_a_c1 0.027855 1.028247 0.018966 1.469 0.141903
cos_turn_a_c2 -0.057819 0.943821 0.018735 -3.086 0.002028
Likelihood ratio test=692.3 on 45 df, p=< 0.00000000000000022
n= 39660, number of events= 3390
(864 observations deleted due to missingness)
$sl_
NULL
$ta_
NULL
$more
NULL
attr(,"class")
[1] "fit_clogit" "list" Code
if(file.exists(paste0("ssf_coefficients/model_id", focal_id, "_3p_harms_wet.rds"))) {
model_3p_harms <- readRDS(paste0("ssf_coefficients/model_id", focal_id, "_3p_harms_wet.rds"))
print("Read existing model")
} else {
tic()
model_3p_harms <- fit_clogit(formula = formula_3p,
data = buffalo_data_scaled_3p)
toc()
# save model object
saveRDS(model_3p_harms, file = paste0("ssf_coefficients/model_id", focal_id, "_3p_harms_wet.rds"))
print("Fitted model")
beep(sound = 2)
}[1] "Read existing model"$model
Call:
survival::clogit(formula, data = data, ...)
coef exp(coef) se(coef) z p
ndvi 0.285568 1.330517 0.100048 2.854 0.00431
ndvi_s1 -1.390180 0.249031 0.331630 -4.192 0.00002765496513804
ndvi_s2 0.306752 1.359004 0.305604 1.004 0.31549
ndvi_s3 0.511608 1.667971 0.300996 1.700 0.08918
ndvi_c1 -0.764679 0.465483 0.293273 -2.607 0.00912
ndvi_c2 -0.278398 0.756995 0.320862 -0.868 0.38558
ndvi_c3 -0.432934 0.648603 0.269812 -1.605 0.10859
ndvi_sq -0.293189 0.745881 0.100592 -2.915 0.00356
ndvi_sq_s1 0.741018 2.098071 0.203963 3.633 0.00028
ndvi_sq_s2 -0.055430 0.946078 0.189001 -0.293 0.76931
ndvi_sq_s3 -0.418150 0.658264 0.189239 -2.210 0.02713
ndvi_sq_c1 0.422817 1.526256 0.185085 2.284 0.02235
ndvi_sq_c2 0.342438 1.408377 0.200928 1.704 0.08833
ndvi_sq_c3 0.196873 1.217589 0.174117 1.131 0.25818
canopy -0.583838 0.557754 0.091736 -6.364 0.00000000019610682
canopy_s1 0.380917 1.463626 0.240952 1.581 0.11390
canopy_s2 -0.197028 0.821167 0.238089 -0.828 0.40793
canopy_s3 -0.052420 0.948930 0.238284 -0.220 0.82588
canopy_c1 -0.073932 0.928734 0.242548 -0.305 0.76051
canopy_c2 0.061447 1.063374 0.245770 0.250 0.80257
canopy_c3 -0.365150 0.694092 0.237933 -1.535 0.12486
canopy_sq 0.478132 1.613059 0.088702 5.390 0.00000007033367032
canopy_sq_s1 0.043557 1.044519 0.165976 0.262 0.79299
canopy_sq_s2 0.124074 1.132100 0.163257 0.760 0.44726
canopy_sq_s3 -0.046423 0.954638 0.163841 -0.283 0.77692
canopy_sq_c1 0.079051 1.082259 0.164240 0.481 0.63030
canopy_sq_c2 0.069921 1.072423 0.167190 0.418 0.67579
canopy_sq_c3 0.153049 1.165382 0.162233 0.943 0.34548
slope -0.006339 0.993681 0.032245 -0.197 0.84415
slope_s1 -0.039502 0.961268 0.041157 -0.960 0.33716
slope_s2 -0.113396 0.892797 0.041491 -2.733 0.00628
slope_s3 0.021763 1.022002 0.041989 0.518 0.60424
slope_c1 -0.024348 0.975946 0.043441 -0.560 0.57515
slope_c2 0.010183 1.010235 0.043000 0.237 0.81280
slope_c3 -0.075361 0.927409 0.041887 -1.799 0.07200
herby 0.024226 1.024522 0.028817 0.841 0.40053
herby_s1 -0.012368 0.987708 0.056910 -0.217 0.82796
herby_s2 -0.026012 0.974323 0.058148 -0.447 0.65463
herby_s3 -0.002061 0.997941 0.058508 -0.035 0.97189
herby_c1 0.055623 1.057199 0.061006 0.912 0.36189
herby_c2 -0.134127 0.874479 0.059428 -2.257 0.02401
herby_c3 0.111934 1.118439 0.056483 1.982 0.04751
step_l -0.572028 0.564380 0.040169 -14.240 < 0.0000000000000002
step_l_s1 0.193827 1.213887 0.042095 4.605 0.00000413339915260
step_l_s2 -0.216874 0.805031 0.042036 -5.159 0.00000024791556794
step_l_s3 0.062735 1.064744 0.043207 1.452 0.14651
step_l_c1 0.132521 1.141703 0.049154 2.696 0.00702
step_l_c2 -0.144700 0.865282 0.044655 -3.240 0.00119
step_l_c3 -0.026849 0.973508 0.042470 -0.632 0.52726
log_step_l 0.630235 1.878051 0.039754 15.853 < 0.0000000000000002
log_step_l_s1 -0.922517 0.397517 0.082748 -11.149 < 0.0000000000000002
log_step_l_s2 0.062840 1.064856 0.065413 0.961 0.33672
log_step_l_s3 0.788434 2.199948 0.065333 12.068 < 0.0000000000000002
log_step_l_c1 -0.349929 0.704738 0.058664 -5.965 0.00000000244639337
log_step_l_c2 -0.653859 0.520035 0.071615 -9.130 < 0.0000000000000002
log_step_l_c3 0.395543 1.485191 0.063387 6.240 0.00000000043725265
cos_turn_a 0.023868 1.024155 0.018997 1.256 0.20898
cos_turn_a_s1 -0.057512 0.944111 0.019158 -3.002 0.00268
cos_turn_a_s2 -0.041156 0.959679 0.018970 -2.170 0.03004
cos_turn_a_s3 0.151176 1.163202 0.019089 7.920 0.00000000000000238
cos_turn_a_c1 0.010286 1.010339 0.019168 0.537 0.59153
cos_turn_a_c2 -0.077024 0.925868 0.019257 -4.000 0.00006339000271146
cos_turn_a_c3 -0.007429 0.992599 0.018872 -0.394 0.69386
Likelihood ratio test=1131 on 63 df, p=< 0.00000000000000022
n= 39660, number of events= 3390
(864 observations deleted due to missingness)
$sl_
NULL
$ta_
NULL
$more
NULL
attr(,"class")
[1] "fit_clogit" "list" Check the fitted model outputs
Create a dataframe of the coefficients with the scaling attributes that we saved when creating the data matrix. We can then return the coefficients to their natural scale by dividing by the scaling factor (standard deviation).
As we can see, we have a coefficient for each covariate by itself, and then one for each of the harmonic interactions. These are the ‘weights’ that we played around with in the Ecography_DynamicSSF_Walkthrough_Harmonics_and_selection_surfaces walkthrough script in: swforrest/dynamic_SSF_sims, and we reconstruct them in exactly the same way. We also have the coefficients for the quadratic terms and the interactions with the harmonics, which we have denoted as ndvi_sq for instance. We will come back to these when we look at the selection surfaces.
$model
Call:
survival::clogit(formula, data = data, ...)
coef exp(coef) se(coef) z p
ndvi 0.31664 1.37250 0.07060 4.485 0.00000729038630065
ndvi_sq -0.26853 0.76450 0.07620 -3.524 0.000425
canopy -0.65865 0.51755 0.08418 -7.824 0.00000000000000512
canopy_sq 0.54151 1.71859 0.08199 6.604 0.00000000003989057
slope 0.01172 1.01179 0.02832 0.414 0.678946
herby 0.04652 1.04762 0.02628 1.771 0.076636
step_l -0.28465 0.75227 0.03014 -9.444 < 0.0000000000000002
log_step_l 0.29081 1.33752 0.02866 10.147 < 0.0000000000000002
cos_turn_a 0.02667 1.02703 0.01814 1.470 0.141504
Likelihood ratio test=229.7 on 9 df, p=< 0.00000000000000022
n= 39660, number of events= 3390
(864 observations deleted due to missingness)
$sl_
NULL
$ta_
NULL
$more
NULL
attr(,"class")
[1] "fit_clogit" "list" Code
# these create massive outputs for the dynamic models so we've commented them out
# model_0p_harms$model$coefficients
# model_0p_harms$se
# model_0p_harms$vcov
# diag(model_0p_harms$D) # between cluster variance
# model_0p_harms$r.effect # individual estimates
# create a dataframe of the coefficients and their scaling attributes
coefs_clr_0p <- data.frame(coefs = names(model_0p_harms$model$coefficients),
value = model_0p_harms$model$coefficients)
coefs_clr_0p$scale_sd <- scaling_attributes_0p$sd
coefs_clr_0p <- coefs_clr_0p %>% mutate(value_nat = value / scale_sd)
head(coefs_clr_0p)Code
# creates a huge output due to the correlation matrix
# model_1p_harms
# model_1p_harms
# model_1p_harms$model$coefficients
# model_1p_harms$se
# model_1p_harms$vcov
# diag(model_1p_harms$D) # between cluster variance
# model_1p_harms$r.effect # individual estimates
coefs_clr_1p <- data.frame(coefs = names(model_1p_harms$model$coefficients),
value = model_1p_harms$model$coefficients)
coefs_clr_1p$scale_sd <- scaling_attributes_1p$sd
coefs_clr_1p <- coefs_clr_1p %>% mutate(value_nat = value / scale_sd)
head(coefs_clr_1p)Code
# creates a huge output due to the correlation matrix
# model_2p_harms
# model_2p_harms
# model_2p_harms$model$coefficients
# model_2p_harms$se
# model_2p_harms$vcov
# diag(model_2p_harms$D) # between cluster variance
# model_2p_harms$r.effect # individual estimates
# creating data frame of model coefficients
coefs_clr_2p <- data.frame(coefs = names(model_2p_harms$model$coefficients),
value = model_2p_harms$model$coefficients)
coefs_clr_2p$scale_sd <- scaling_attributes_2p$sd
coefs_clr_2p <- coefs_clr_2p %>% mutate(value_nat = value / scale_sd)
head(coefs_clr_2p)Code
# creates a huge output due to the correlation matrix
# model_3p_harms
# model_3p_harms$model$coefficients
# model_3p_harms$se
# model_3p_harms$vcov
# diag(model_3p_harms$D) # between cluster variance
# model_3p_harms$r.effect # individual estimates
# creating dataframe of coefficients
coefs_clr_3p <- data.frame(coefs = names(model_3p_harms$model$coefficients),
value = model_3p_harms$model$coefficients)
coefs_clr_3p$scale_sd <- scaling_attributes_3p$sd
coefs_clr_3p <- coefs_clr_3p %>% mutate(value_nat = value / scale_sd)
head(coefs_clr_3p)Reconstruct the temporally dynamic coefficients
First we reconstruct the hourly coefficients for the model with no harmonics. This step isn’t necessary as we already have the coefficients, and we have already rescaled them in the dataframe we created above. But as we are also fitting harmonic models and recover their coefficients across time, we have used the same approach here so then we can plot them together and illustrate the static/dynamic outputs of the models. It also means that we can use the same simulation code (which indexes across the hour of the day), and just change the data frame of coefficients (as it will index across the coefficients of the static model but they won’t change).
We need a sequence of values that covers a full period (or the period that we want to construct the function over, which can be more or less than 1 period). The sequence can be arbitrarily finely spaced. The smaller the increment the smoother the function will be for plotting. When simulating data from the temporally dynamic coefficients, we will subset to the increment that relates to the data collection and model fitting (i.e. one hour in this case).
Essentially, the coefficients can be considered as weights of the harmonics, which combine into a single function.
Now we can reconstruct the harmonic function using the formula that we put into our model by interacting the harmonic terms with each of the covariates, for two pairs of harmonics (2p) a single covariate, let’s say herbaceous vegetation (herby), this would be written down as:
\[ f = \beta_{herby} + \beta_{herby\_s1} \sin\left(\frac{2\pi t}{24}\right) + \beta_{herby\_c1} \cos\left(\frac{2\pi t}{24}\right) + \beta_{herby\_s2} \sin\left(\frac{4\pi t}{24}\right) + \beta_{herby\_c2} \cos\left(\frac{4\pi t}{24}\right), \]
where we have 5 \(\beta_{herby}\) coefficients, one for the linear term, and one for each of the harmonic terms.
Here we use matrix multiplication to reconstruct the temporally dynamic coefficients. We provide some background in the Ecography_DynamicSSF_Walkthrough_Harmonics_and_selection_surfaces script.
First we create a matrix of the values of the harmonics, which is just the sin and cos terms for each harmonic, and then we can multiply this by the coefficients to get the function. When we use two pairs of harmonics we will have 5 coefficients for each covariate (linear + 2 sine and 2 cosine), so there will be 5 columns in the matrix.
For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the second matrix. The result will then have the same number of rows as the first matrix and the same number of columns as the second matrix.
Or in other words, if we have a 24 x 5 matrix of harmonics and a 5 x 1 matrix of coefficients, we will get a 24 x 1 matrix of the function, which corresponds to our 24 hours of the day.
Code
# increments are arbitrary - finer results in smoother curves
# for the simulations we will subset to the step interval
hour <- seq(0,23.9,0.1)
# create the dataframe of values of the harmonic terms over the period (here just the linear term)
hour_harmonics_df_0p <- data.frame("linear_term" = rep(1, length(hour)))
harmonics_scaled_df_0p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"ndvi_2" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"canopy" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"canopy_2" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"slope" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("slope", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"herby" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"sl" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"log_sl" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"cos_ta" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))))
harmonics_scaled_long_0p <- pivot_longer(harmonics_scaled_df_0p,
cols = !1,
names_to = "coef")Code
# create the dataframe of values of the harmonic terms over the period
hour_harmonics_df_1p <- data.frame("linear_term" = rep(1, length(hour)),
"hour_s1" = sin(2*pi*hour/24),
"hour_c1" = cos(2*pi*hour/24))
harmonics_scaled_df_1p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"ndvi_2" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"canopy" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"canopy_2" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"slope" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("slope", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"herby" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"sl" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"log_sl" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"cos_ta" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))))
harmonics_scaled_long_1p <- pivot_longer(harmonics_scaled_df_1p,
cols = !1,
names_to = "coef")Code
# create the dataframe of values of the harmonic terms over the period
hour_harmonics_df_2p <- data.frame("linear_term" = rep(1, length(hour)),
"hour_s1" = sin(2*pi*hour/24),
"hour_s2" = sin(4*pi*hour/24),
"hour_c1" = cos(2*pi*hour/24),
"hour_c2" = cos(4*pi*hour/24))
harmonics_scaled_df_2p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"ndvi_2" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"canopy" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"canopy_2" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"slope" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("slope", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"herby" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"sl" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"log_sl" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"cos_ta" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))))
harmonics_scaled_long_2p <- pivot_longer(harmonics_scaled_df_2p, cols = !1,
names_to = "coef")Code
# create the dataframe of values of the harmonic terms over the period
hour_harmonics_df_3p <- data.frame("linear_term" = rep(1, length(hour)),
"hour_s1" = sin(2*pi*hour/24),
"hour_s2" = sin(4*pi*hour/24),
"hour_s3" = sin(6*pi*hour/24),
"hour_c1" = cos(2*pi*hour/24),
"hour_c2" = cos(4*pi*hour/24),
"hour_c3" = cos(6*pi*hour/24))
harmonics_scaled_df_3p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"ndvi_2" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"canopy" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"canopy_2" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"slope" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("slope", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"herby" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"sl" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"log_sl" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"cos_ta" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))))
harmonics_scaled_long_3p <- pivot_longer(harmonics_scaled_df_3p, cols = !1,
names_to = "coef")Plot the results - scaled temporally dynamic coefficients
Here we show the temporally-varying coefficients across time (which are currently still scaled).
Code
ggplot() +
geom_path(data = harmonics_scaled_long_0p,
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour") +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Code
ggplot() +
geom_path(data = harmonics_scaled_long_1p,
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour") +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Code
ggplot() +
geom_path(data = harmonics_scaled_long_2p,
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour") +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Code
ggplot() +
geom_path(data = harmonics_scaled_long_3p,
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour") +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Reconstructing the natural-scale temporally dynamic coefficients
As we scaled the covariate values prior to fitting the models, we want to rescale the coefficients to their natural scale. This is important for the simulations, as the environmental variables will not be scaled when we simulate steps.
Code
harmonics_nat_df_0p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"ndvi_2" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"canopy" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"canopy_2" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"slope" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("slope", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"herby" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"sl" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"log_sl" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))),
"cos_ta" = as.numeric(
coefs_clr_0p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_0p))))Code
harmonics_nat_df_1p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"ndvi_2" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"canopy" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"canopy_2" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"slope" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("slope", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"herby" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"sl" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"log_sl" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))),
"cos_ta" = as.numeric(
coefs_clr_1p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_1p))))Code
harmonics_nat_df_2p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"ndvi_2" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"canopy" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"canopy_2" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"slope" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("slope", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"herby" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"sl" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"log_sl" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))),
"cos_ta" = as.numeric(
coefs_clr_2p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_2p))))Code
harmonics_nat_df_3p <- data.frame(
"hour" = hour,
"ndvi" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("ndvi", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"ndvi_2" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("ndvi_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"canopy" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("canopy", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"canopy_2" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("canopy_sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"slope" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("slope", coefs) & !grepl("sq", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"herby" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("herby", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"sl" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("step_l", coefs) & !grepl("log", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"log_sl" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("log_step_l", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))),
"cos_ta" = as.numeric(
coefs_clr_3p %>% dplyr::filter(grepl("cos", coefs)) %>%
pull(value_nat) %>% t() %*% t(as.matrix(hour_harmonics_df_3p))))Update the Gamma and von Mises distributions
To update the Gamma and von Mises distribution from the tentative distributions (e.g. Fieberg et al. 2021, Appendix C), we just do the calculation at each time point (for the natural-scale coefficients).
Code
# from the step generation script
tentative_shape <- 0.438167
tentative_scale <- 534.3507
tentative_kappa <- 0.1848126
hour_coefs_nat_df_0p <- harmonics_nat_df_0p %>%
mutate(shape = tentative_shape + log_sl,
scale = 1/((1/tentative_scale) - sl),
kappa = tentative_kappa + cos_ta)
# save the coefficients to use in the simulations
write_csv(hour_coefs_nat_df_0p,
paste0("ssf_coefficients/id", focal_id, "_0pDaily_coefs_wet_", Sys.Date(), ".csv"))
# turning into a long data frame
hour_coefs_nat_long_0p <- pivot_longer(hour_coefs_nat_df_0p,
cols = !1,
names_to = "coef")Code
hour_coefs_nat_df_1p <- harmonics_nat_df_1p %>%
mutate(shape = tentative_shape + log_sl,
scale = 1/((1/tentative_scale) - sl),
kappa = tentative_kappa + cos_ta)
# save the coefficients to use in the simulations
write_csv(hour_coefs_nat_df_1p,
paste0("ssf_coefficients/id", focal_id, "_1pDaily_coefs_wet_",Sys.Date(), ".csv"))
# turning into a long data frame
hour_coefs_nat_long_1p <- pivot_longer(hour_coefs_nat_df_1p,
cols = !1, names_to = "coef")Code
hour_coefs_nat_df_2p <- harmonics_nat_df_2p %>%
mutate(shape = tentative_shape + log_sl,
scale = 1/((1/tentative_scale) - sl),
kappa = tentative_kappa + cos_ta)
# save the coefficients to use in the simulations
write_csv(hour_coefs_nat_df_2p,
paste0("ssf_coefficients/id", focal_id, "_2pDaily_coefs_wet_",Sys.Date(), ".csv"))
# turning into a long data frame
hour_coefs_nat_long_2p <- pivot_longer(hour_coefs_nat_df_2p, cols = !1,
names_to = "coef")Code
hour_coefs_nat_df_3p <- harmonics_nat_df_3p %>%
mutate(shape = tentative_shape + log_sl,
scale = 1/((1/tentative_scale) - sl),
kappa = tentative_kappa + cos_ta)
# save the coefficients to use in the simulations
write_csv(hour_coefs_nat_df_3p,
paste0("ssf_coefficients/id", focal_id, "_3pDaily_coefs_wet_", Sys.Date(), ".csv"))
# turning into a long data frame
hour_coefs_nat_long_3p <- pivot_longer(hour_coefs_nat_df_3p, cols = !1,
names_to = "coef")Plot the natural-scale temporally dynamic coefficients
Now that the coefficients are in their natural scales, they will be larger or smaller depending on the scale of the covariate.
Plot just the habitat selection coefficients.
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_0p %>%
filter(!coef %in% c("shape", "scale", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_1p %>%
filter(!coef %in% c("shape", "scale", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_2p %>%
filter(!coef %in% c("shape", "scale", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_3p %>%
filter(!coef %in% c("shape", "scale", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(Time-varying~parameter~values~beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_color_discrete("Estimate") +
theme_classic() +
theme(legend.position = "bottom")
Plot only the temporally dynamic movement parameters
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_0p %>%
filter(coef %in% c("shape", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_path(data = hour_coefs_nat_long_0p %>%
filter(coef == "scale"),
aes(x = hour, y = value/1000, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
ggtitle("Note that the scale parameter is divided by 1000 for plotting") +
scale_color_discrete("Estimate",
labels = c("kappa" = "von Mises kappa",
"scale" = "Gamma scale / 1000",
"shape" = "Gamma shape")) +
theme_classic() +
theme(legend.position = "right")
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_1p %>%
filter(coef %in% c("shape", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_path(data = hour_coefs_nat_long_1p %>%
filter(coef == "scale"),
aes(x = hour, y = value/1000, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
ggtitle("Note that the scale parameter is divided by 1000 for plotting") +
scale_color_discrete("Estimate",
labels = c("kappa" = "von Mises kappa",
"scale" = "Gamma scale / 1000",
"shape" = "Gamma shape")) +
theme_classic() +
theme(legend.position = "right")
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_2p %>%
filter(coef %in% c("shape", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_path(data = hour_coefs_nat_long_2p %>%
filter(coef == "scale"),
aes(x = hour, y = value/1000, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous("Value of parameter") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
ggtitle("*Note that the scale parameter is divided by 1000 for plotting") +
scale_color_discrete("Estimate",
labels = c("kappa" = "von Mises kappa",
"scale" = "Gamma scale / 1000",
"shape" = "Gamma shape")) +
theme_classic() +
theme(legend.position = "right")
Code
ggplot() +
geom_path(data = hour_coefs_nat_long_3p %>%
filter(coef %in% c("shape", "kappa")),
aes(x = hour, y = value, colour = coef)) +
geom_path(data = hour_coefs_nat_long_3p %>%
filter(coef == "scale"),
aes(x = hour, y = value/1000, colour = coef)) +
geom_hline(yintercept = 0, linetype = "dashed") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
ggtitle("Note that the scale parameter is divided by 1000 for plotting") +
scale_color_discrete("Estimate",
labels = c("kappa" = "von Mises kappa",
"scale" = "Gamma scale / 1000",
"shape" = "Gamma shape")) +
theme_classic() +
theme(legend.position = "right")
Sample from temporally dynamic movement parameters
Here we sample from the movement kernel to generate a distribution of step lengths for each hour of the day, to assess how well it matches the observed step lengths. This is the ‘selection-free’ movement kernel, so the step lengths and turning angles from the simulations will be different, as the steps will be conditioned on the habitat, but this is a useful diagnostic to assess whether the harmonics are capturing the observed movement dynamics.
Code
`summarise()` has grouped output by 'id'. You can override using the `.groups`
argument.Code
# number of samples at each hour (more = smoother plotting, but slower)
n <- 1e5
gamma_dist_list <- vector(mode = "list", length = nrow(hour_coefs_nat_df_0p))
gamma_mean <- c()
gamma_median <- c()
gamma_ratio <- c()
for(hour_no in 1:nrow(hour_coefs_nat_df_0p)) {
gamma_dist_list[[hour_no]] <- rgamma(n, shape = hour_coefs_nat_df_0p$shape[hour_no],
scale = hour_coefs_nat_df_0p$scale[hour_no])
gamma_mean[hour_no] <- mean(gamma_dist_list[[hour_no]])
gamma_median[hour_no] <- median(gamma_dist_list[[hour_no]])
gamma_ratio[hour_no] <- gamma_mean[hour_no] / gamma_median[hour_no]
}
gamma_df_0p <- data.frame(model = "0p",
hour = hour_coefs_nat_df_0p$hour,
mean = gamma_mean,
median = gamma_median,
ratio = gamma_ratio)
mean_sl_0p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = mean_sl, colour = factor(id))) +
geom_path(data = gamma_df_0p,
aes(x = hour, y = mean), colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Mean step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled mean step length",
subtitle = "No harmonics") +
theme_classic() +
theme(legend.position = "right")
mean_sl_0p
Code
median_sl_0p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = median_sl, colour = factor(id))) +
geom_path(data = gamma_df_0p, aes(x = hour, y = median),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Median step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled median step length",
subtitle = "No harmonics") +
theme_classic() +
theme(legend.position = "right")
median_sl_0p
Code
Code
Code
Code
gamma_dist_list <- vector(mode = "list", length = nrow(hour_coefs_nat_df_1p))
gamma_mean <- c()
gamma_median <- c()
gamma_ratio <- c()
for(hour_no in 1:nrow(hour_coefs_nat_df_1p)) {
gamma_dist_list[[hour_no]] <- rgamma(n,
shape = hour_coefs_nat_df_1p$shape[hour_no],
scale = hour_coefs_nat_df_1p$scale[hour_no])
gamma_mean[hour_no] <- mean(gamma_dist_list[[hour_no]])
gamma_median[hour_no] <- median(gamma_dist_list[[hour_no]])
gamma_ratio[hour_no] <- gamma_mean[hour_no] / gamma_median[hour_no]
}
gamma_df_1p <- data.frame(model = "1p",
hour = hour_coefs_nat_df_1p$hour,
mean = gamma_mean,
median = gamma_median,
ratio = gamma_ratio)
mean_sl_1p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = mean_sl, colour = factor(id))) +
geom_path(data = gamma_df_1p,
aes(x = hour, y = mean),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Mean step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled mean step length",
subtitle = "One pair of harmonics") +
theme_classic() +
theme(legend.position = "none")
mean_sl_1p
Code
median_sl_1p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = median_sl, colour = factor(id))) +
geom_path(data = gamma_df_1p,
aes(x = hour, y = median),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Median step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled median step length",
subtitle = "One pair of harmonics") +
theme_classic() +
theme(legend.position = "none")
median_sl_1p
Code
Code
Code
Code
gamma_dist_list <- vector(mode = "list", length = nrow(hour_coefs_nat_df_2p))
gamma_mean <- c()
gamma_median <- c()
gamma_ratio <- c()
for(hour_no in 1:nrow(hour_coefs_nat_df_2p)) {
gamma_dist_list[[hour_no]] <- rgamma(n,
shape = hour_coefs_nat_df_2p$shape[hour_no],
scale = hour_coefs_nat_df_2p$scale[hour_no])
gamma_mean[hour_no] <- mean(gamma_dist_list[[hour_no]])
gamma_median[hour_no] <- median(gamma_dist_list[[hour_no]])
gamma_ratio[hour_no] <- gamma_mean[hour_no] / gamma_median[hour_no]
}
gamma_df_2p <- data.frame(model = "2p",
hour = hour_coefs_nat_df_2p$hour,
mean = gamma_mean,
median = gamma_median,
ratio = gamma_ratio)
mean_sl_2p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = mean_sl, colour = factor(id))) +
geom_path(data = gamma_df_2p,
aes(x = hour, y = mean),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Mean step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled mean step length",
subtitle = "Two pairs of harmonics") +
theme_classic() +
theme(legend.position = "none")
mean_sl_2p
Code
median_sl_2p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = median_sl, colour = factor(id))) +
geom_path(data = gamma_df_2p,
aes(x = hour, y = median),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Median step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled median step length",
subtitle = "Two pairs of harmonics") +
theme_classic() +
theme(legend.position = "none")
median_sl_2p
Code
Code
Code
Code
gamma_dist_list <- vector(mode = "list", length = nrow(hour_coefs_nat_df_3p))
gamma_mean <- c()
gamma_median <- c()
gamma_ratio <- c()
for(hour_no in 1:nrow(hour_coefs_nat_df_3p)) {
gamma_dist_list[[hour_no]] <- rgamma(n,
shape = hour_coefs_nat_df_3p$shape[hour_no],
scale = hour_coefs_nat_df_3p$scale[hour_no])
gamma_mean[hour_no] <- mean(gamma_dist_list[[hour_no]])
gamma_median[hour_no] <- median(gamma_dist_list[[hour_no]])
gamma_ratio[hour_no] <- gamma_mean[hour_no] / gamma_median[hour_no]
}
gamma_df_3p <- data.frame(model = "3p",
hour = hour_coefs_nat_df_3p$hour,
mean = gamma_mean,
median = gamma_median,
ratio = gamma_ratio)
mean_sl_3p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = mean_sl, colour = factor(id))) +
geom_path(data = gamma_df_3p,
aes(x = hour, y = mean),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Mean step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled mean step length",
subtitle = "Three pairs of harmonics") +
theme_classic() +
theme(legend.position = "none")
mean_sl_3p
Code
median_sl_3p <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = median_sl, colour = factor(id))) +
geom_path(data = gamma_df_3p,
aes(x = hour, y = median),
colour = "red", linetype = "dashed") +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Median step length") +
scale_colour_viridis_d("Buffalo") +
ggtitle("Observed and modelled median step length",
subtitle = "Three pairs of harmonics") +
theme_classic() +
theme(legend.position = "none")
median_sl_3p
Code
Code
Code
Creating selection surfaces
As we have both quadratic and harmonic terms in the model, we can reconstruct a ‘selection surface’ to visualise how the animal’s respond to environmental features changes through time.
To illustrate, if we don’t have temporal dynamics (as is the case for this model), then we have a coefficient for the linear term and a coefficient for the quadratic term. Using these, we can plot the selection curve at the scale of the environmental variable (in this case NDVI).
Using the natural scale coefficients from the model:
Code
# first get a sequence of NDVI values,
# starting from the minimum observed in the data to the maximum
ndvi_min <- min(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_max <- max(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_seq <- seq(ndvi_min, ndvi_max, by = 0.01)
# take the coefficients from the model and calculation the selection value
# for every NDVI value in this sequence
# we can separate to the linear term
ndvi_linear_selection <- hour_coefs_nat_df_0p$ndvi[1] * ndvi_seq
plot(x = ndvi_seq, y = ndvi_linear_selection,
main = "Selection for NDVI - linear term",
xlab = "NDVI", ylab = "Estimated selection")
lines(ndvi_seq, rep(0,length(ndvi_seq)), lty = "dashed")
Code
Code
# and the sum of both
ndvi_sum_selection <- ndvi_linear_selection + ndvi_quadratic_selection
plot(x = ndvi_seq, y = ndvi_sum_selection,
main = "Selection for NDVI - sum of linear and quadratic terms",
xlab = "NDVI", ylab = "Estimated selection")
lines(ndvi_seq, rep(0,length(ndvi_seq)), lty = "dashed")
When there are no temporal dynamics, then this quadratic curve will be the same throughout the day, but when we have temporally dynamic coefficients for both the linear term and the quadratic term, then we will have a curves that vary continuously throughout the day, which we can visualise as a selection surface.
Here we illustrate for the model with 2 pairs of harmonic terms.
For brevity we won’t plot the linear and quadratic terms separately, but we can do so if needed.
First for Hour 3
Code
hour_no <- 3
# we can separate to the linear term
ndvi_linear_selection <-
hour_coefs_nat_df_1p$ndvi[which(hour_coefs_nat_df_1p$hour == hour_no)] * ndvi_seq
# plot(x = ndvi_seq, y = ndvi_linear_selection,
# main = "Selection for NDVI - linear term",
# xlab = "NDVI", ylab = "Estimated selection")
# and the quadratic term
ndvi_quadratic_selection <-
(hour_coefs_nat_df_1p$ndvi_2[which(hour_coefs_nat_df_1p$hour == hour_no)] * (ndvi_seq ^ 2))
# plot(x = ndvi_seq, y = ndvi_quadratic_selection,
# main = "Selection for NDVI - quadratic term",
# xlab = "NDVI", ylab = "Estimated selection")
# and the sum of both
ndvi_sum_selection <- ndvi_linear_selection + ndvi_quadratic_selection
plot(x = ndvi_seq, y = ndvi_sum_selection,
main = "Selection for NDVI - sum of linear and quadratic terms",
xlab = "NDVI", ylab = "Estimated selection")
lines(ndvi_seq, rep(0,length(ndvi_seq)), lty = "dashed")
We can see that the coefficient at hour 3 shows highest selection for NDVI values slightly above 0.2, and the coefficient is mostly negative.
Secondly for Hour 12
Code
hour_no <- 12
# we can separate to the linear term
ndvi_linear_selection <-
hour_coefs_nat_df_1p$ndvi[which(hour_coefs_nat_df_1p$hour == hour_no)] * ndvi_seq
# plot(x = ndvi_seq, y = ndvi_linear_selection,
# main = "Selection for NDVI - linear term",
# xlab = "NDVI", ylab = "Estimated selection")
# and the quadratic term
ndvi_quadratic_selection <-
(hour_coefs_nat_df_1p$ndvi_2[which(hour_coefs_nat_df_1p$hour == hour_no)] * (ndvi_seq ^ 2))
# plot(x = ndvi_seq, y = ndvi_quadratic_selection,
# main = "Selection for NDVI - quadratic term",
# xlab = "NDVI", ylab = "Estimated selection")
# and the sum of both
ndvi_sum_selection <- ndvi_linear_selection + ndvi_quadratic_selection
plot(x = ndvi_seq, y = ndvi_sum_selection,
main = "Selection for NDVI - sum of linear and quadratic terms",
xlab = "NDVI", ylab = "Estimated selection")
lines(ndvi_seq, rep(0,length(ndvi_seq)), lty = "dashed")
Whereas for hour 12, the coefficient shows highest selection for NDVI values slightly above 0.4, and the coefficient is positive for NDVI values above 0.
We can imagine viewing these plots for every hour of the day, where each hour has a different quadratic curve, but this would be a lot of plots. We can also see it as a 3D surface, where the x-axis is the hour of the day, the y-axis is the NDVI value, and the z-axis (colour) is the coefficient value.
We simply index over the linear and quadratic terms and calculate the coefficient values at every time point.
NDVI selection surface
Code
ndvi_min <- min(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_max <- max(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_seq <- seq(ndvi_min, ndvi_max, by = 0.01)
# Create empty data frame
ndvi_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_0p),
nrow = length(ndvi_seq)))
# loop over each time increment, calculating the selection values for each NDVI value
# and storing each time increment as a column in a dataframe that we can use for plotting
for(i in 1:nrow(hour_coefs_nat_df_0p)) {
# Assign the vector as a column to the dataframe
ndvi_fresponse_df[,i] <- (hour_coefs_nat_df_0p$ndvi[i] * ndvi_seq) +
(hour_coefs_nat_df_0p$ndvi_2[i] * (ndvi_seq ^ 2))
}
ndvi_fresponse_df <- data.frame(ndvi_seq, ndvi_fresponse_df)
colnames(ndvi_fresponse_df) <- c("ndvi", hour)
ndvi_fresponse_long <- pivot_longer(ndvi_fresponse_df,
cols = !1, names_to = "hour")
ndvi_contour_max <- max(ndvi_fresponse_long$value) # 0.7890195
ndvi_contour_min <- min(ndvi_fresponse_long$value) # -0.7945691
ndvi_contour_increment <- (ndvi_contour_max-ndvi_contour_min)/10
ndvi_quad_0p <- ggplot(data = ndvi_fresponse_long,
aes(x = as.numeric(hour), y = ndvi)) +
geom_point(aes(colour = value)) +
geom_contour(aes(z = value),
breaks = seq(ndvi_contour_increment,
ndvi_contour_max,
ndvi_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-ndvi_contour_increment,
ndvi_contour_min,
-ndvi_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("NDVI value", breaks = seq(-1, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
ggtitle("Normalised Difference Vegetation Index (NDVI)") +
theme_classic() +
theme(legend.position = "none")
ndvi_quad_0p
Code
ndvi_min <- min(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_max <- max(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_seq <- seq(ndvi_min, ndvi_max, by = 0.01)
# Create empty data frame
ndvi_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_1p),
nrow = length(ndvi_seq)))
for(i in 1:nrow(hour_coefs_nat_df_1p)) {
# Assign the vector as a column to the dataframe
ndvi_fresponse_df[,i] <- (hour_coefs_nat_df_1p$ndvi[i] * ndvi_seq) +
(hour_coefs_nat_df_1p$ndvi_2[i] * (ndvi_seq ^ 2))
}
ndvi_fresponse_df <- data.frame(ndvi_seq, ndvi_fresponse_df)
colnames(ndvi_fresponse_df) <- c("ndvi", hour)
ndvi_fresponse_long <- pivot_longer(ndvi_fresponse_df, cols = !1, names_to = "hour")
ndvi_contour_max <- max(ndvi_fresponse_long$value) # 0.7890195
ndvi_contour_min <- min(ndvi_fresponse_long$value) # -0.7945691
ndvi_contour_increment <- (ndvi_contour_max-ndvi_contour_min)/10
ndvi_quad_1p <- ggplot(data = ndvi_fresponse_long,
aes(x = as.numeric(hour), y = ndvi)) +
geom_point(aes(colour = value)) + # colour = "white"
geom_contour(aes(z = value),
breaks = seq(ndvi_contour_increment,
ndvi_contour_max,
ndvi_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-ndvi_contour_increment,
ndvi_contour_min,
-ndvi_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("NDVI value", breaks = seq(-1, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
# ggtitle("Normalised Difference Vegetation Index (NDVI)") +
theme_classic() +
theme(legend.position = "none")
ndvi_quad_1p
Code
ndvi_min <- min(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_max <- max(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_seq <- seq(ndvi_min, ndvi_max, by = 0.01)
# Create empty data frame
ndvi_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_2p),
nrow = length(ndvi_seq)))
for(i in 1:nrow(hour_coefs_nat_df_2p)) {
# Assign the vector as a column to the dataframe
ndvi_fresponse_df[,i] <- (hour_coefs_nat_df_2p$ndvi[i] * ndvi_seq) +
(hour_coefs_nat_df_2p$ndvi_2[i] * (ndvi_seq ^ 2))
}
ndvi_fresponse_df <- data.frame(ndvi_seq, ndvi_fresponse_df)
colnames(ndvi_fresponse_df) <- c("ndvi", hour)
ndvi_fresponse_long <- pivot_longer(ndvi_fresponse_df, cols = !1,
names_to = "hour")
ndvi_contour_max <- max(ndvi_fresponse_long$value) # 0.7890195
ndvi_contour_min <- min(ndvi_fresponse_long$value) # -0.7945691
ndvi_contour_increment <- (ndvi_contour_max-ndvi_contour_min)/10
ndvi_quad_2p <- ggplot(data = ndvi_fresponse_long,
aes(x = as.numeric(hour), y = ndvi)) +
geom_point(aes(colour = value)) + # colour = "white"
geom_contour(aes(z = value),
breaks = seq(ndvi_contour_increment,
ndvi_contour_max,
ndvi_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-ndvi_contour_increment,
ndvi_contour_min,
-ndvi_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("NDVI value", breaks = seq(-1, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
# ggtitle("Normalised Difference Vegetation Index (NDVI)") +
theme_classic() +
theme(legend.position = "right")
ndvi_quad_2p
Code
ndvi_min <- min(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_max <- max(buffalo_data$ndvi_temporal, na.rm = TRUE)
ndvi_seq <- seq(ndvi_min, ndvi_max, by = 0.01)
# Create empty data frame
ndvi_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_3p),
nrow = length(ndvi_seq)))
for(i in 1:nrow(hour_coefs_nat_df_3p)) {
# Assign the vector as a column to the dataframe
ndvi_fresponse_df[,i] <- (hour_coefs_nat_df_3p$ndvi[i] * ndvi_seq) +
(hour_coefs_nat_df_3p$ndvi_2[i] * (ndvi_seq ^ 2))
}
ndvi_fresponse_df <- data.frame(ndvi_seq, ndvi_fresponse_df)
colnames(ndvi_fresponse_df) <- c("ndvi", hour)
ndvi_fresponse_long <- pivot_longer(ndvi_fresponse_df, cols = !1,
names_to = "hour")
ndvi_contour_max <- max(ndvi_fresponse_long$value) # 0.7890195
ndvi_contour_min <- min(ndvi_fresponse_long$value) # -0.7945691
ndvi_contour_increment <- (ndvi_contour_max-ndvi_contour_min)/10
ndvi_quad_3p <- ggplot(data = ndvi_fresponse_long,
aes(x = as.numeric(hour), y = ndvi)) +
geom_point(aes(colour = value)) + # colour = "white"
geom_contour(aes(z = value),
breaks = seq(ndvi_contour_increment,
ndvi_contour_max,
ndvi_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-ndvi_contour_increment,
ndvi_contour_min,
-ndvi_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("NDVI value", breaks = seq(-1, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
# ggtitle("Normalised Difference Vegetation Index (NDVI)") +
theme_classic() +
theme(legend.position = "none")
ndvi_quad_3p
Canopy cover selection surface
Code
canopy_min <- min(buffalo_data$canopy_01, na.rm = TRUE)
canopy_max <- max(buffalo_data$canopy_01, na.rm = TRUE)
canopy_seq <- seq(canopy_min, canopy_max, by = 0.01)
# Create empty data frame
canopy_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_0p),
nrow = length(canopy_seq)))
for(i in 1:nrow(hour_coefs_nat_df_0p)) {
# Assign the vector as a column to the dataframe
canopy_fresponse_df[,i] <- (hour_coefs_nat_df_0p$canopy[i] * canopy_seq) +
(hour_coefs_nat_df_0p$canopy_2[i] * (canopy_seq ^ 2))
}
canopy_fresponse_df <- data.frame(canopy_seq, canopy_fresponse_df)
colnames(canopy_fresponse_df) <- c("canopy", hour)
canopy_fresponse_long <- pivot_longer(canopy_fresponse_df,
cols = !1,
names_to = "hour")
canopy_contour_min <- min(canopy_fresponse_long$value) # 0
canopy_contour_max <- max(canopy_fresponse_long$value) # 2.181749
canopy_contour_increment <- (canopy_contour_max-canopy_contour_min)/10
canopy_quad_0p <- ggplot(data = canopy_fresponse_long, aes(x = as.numeric(hour),
y = canopy)) +
geom_point(aes(colour = value)) +
geom_contour(aes(z = value),
breaks = seq(canopy_contour_increment, canopy_contour_max,
-canopy_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-canopy_contour_increment, canopy_contour_min,
-canopy_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("Canopy cover", breaks = seq(0, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
ggtitle("Canopy Cover") +
theme_classic() +
theme(legend.position = "none")
canopy_quad_0p
Code
canopy_min <- min(buffalo_data$canopy_01, na.rm = TRUE)
canopy_max <- max(buffalo_data$canopy_01, na.rm = TRUE)
canopy_seq <- seq(canopy_min, canopy_max, by = 0.01)
# Create empty data frame
canopy_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_1p),
nrow = length(canopy_seq)))
for(i in 1:nrow(hour_coefs_nat_df_1p)) {
# Assign the vector as a column to the dataframe
canopy_fresponse_df[,i] <- (hour_coefs_nat_df_1p$canopy[i] * canopy_seq) +
(hour_coefs_nat_df_1p$canopy_2[i] * (canopy_seq ^ 2))
}
canopy_fresponse_df <- data.frame(canopy_seq, canopy_fresponse_df)
colnames(canopy_fresponse_df) <- c("canopy", hour)
canopy_fresponse_long <- pivot_longer(canopy_fresponse_df, cols = !1,
names_to = "hour")
canopy_contour_min <- min(canopy_fresponse_long$value) # 0
canopy_contour_max <- max(canopy_fresponse_long$value) # 2.181749
canopy_contour_increment <- (canopy_contour_max-canopy_contour_min)/10
canopy_quad_1p <- ggplot(data = canopy_fresponse_long,
aes(x = as.numeric(hour), y = canopy)) +
geom_point(aes(colour = value)) +
geom_contour(aes(z = value),
breaks = seq(canopy_contour_increment,
canopy_contour_max,
canopy_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-canopy_contour_increment,
canopy_contour_min,
-canopy_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("Canopy cover", breaks = seq(0, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
# ggtitle("Canopy Cover") +
theme_classic() +
theme(legend.position = "none")
canopy_quad_1p
Code
canopy_min <- min(buffalo_data$canopy_01, na.rm = TRUE)
canopy_max <- max(buffalo_data$canopy_01, na.rm = TRUE)
canopy_seq <- seq(canopy_min, canopy_max, by = 0.01)
# Create empty data frame
canopy_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_2p),
nrow = length(canopy_seq)))
for(i in 1:nrow(hour_coefs_nat_df_2p)) {
# Assign the vector as a column to the dataframe
canopy_fresponse_df[,i] <- (hour_coefs_nat_df_2p$canopy[i] * canopy_seq) +
(hour_coefs_nat_df_2p$canopy_2[i] * (canopy_seq ^ 2))
}
canopy_fresponse_df <- data.frame(canopy_seq, canopy_fresponse_df)
colnames(canopy_fresponse_df) <- c("canopy", hour)
canopy_fresponse_long <- pivot_longer(canopy_fresponse_df, cols = !1,
names_to = "hour")
canopy_contour_min <- min(canopy_fresponse_long$value) # 0
canopy_contour_max <- max(canopy_fresponse_long$value) # 2.181749
canopy_contour_increment <- (canopy_contour_max-canopy_contour_min)/10
canopy_quad_2p <- ggplot(data = canopy_fresponse_long,
aes(x = as.numeric(hour), y = canopy)) +
geom_point(aes(colour = value)) +
geom_contour(aes(z = value),
breaks = seq(canopy_contour_increment,
canopy_contour_max,
canopy_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-canopy_contour_increment,
canopy_contour_min,
-canopy_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("Canopy cover", breaks = seq(0, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
# ggtitle("Canopy Cover") +
theme_classic() +
theme(legend.position = "none")
canopy_quad_2p
Code
canopy_min <- min(buffalo_data$canopy_01, na.rm = TRUE)
canopy_max <- max(buffalo_data$canopy_01, na.rm = TRUE)
canopy_seq <- seq(canopy_min, canopy_max, by = 0.01)
# Create empty data frame
canopy_fresponse_df <- data.frame(matrix(ncol = nrow(hour_coefs_nat_df_3p),
nrow = length(canopy_seq)))
for(i in 1:nrow(hour_coefs_nat_df_3p)) {
# Assign the vector as a column to the dataframe
canopy_fresponse_df[,i] <- (hour_coefs_nat_df_3p$canopy[i] * canopy_seq) +
(hour_coefs_nat_df_3p$canopy_2[i] * (canopy_seq ^ 2))
}
canopy_fresponse_df <- data.frame(canopy_seq, canopy_fresponse_df)
colnames(canopy_fresponse_df) <- c("canopy", hour)
canopy_fresponse_long <- pivot_longer(canopy_fresponse_df, cols = !1,
names_to = "hour")
canopy_contour_min <- min(canopy_fresponse_long$value) # 0
canopy_contour_max <- max(canopy_fresponse_long$value) # 2.181749
canopy_contour_increment <- (canopy_contour_max-canopy_contour_min)/10
canopy_quad_3p <- ggplot(data = canopy_fresponse_long,
aes(x = as.numeric(hour), y = canopy)) +
geom_point(aes(colour = value)) +
geom_contour(aes(z = value),
breaks = seq(canopy_contour_increment,
canopy_contour_max,
canopy_contour_increment),
colour = "black", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value),
breaks = seq(-canopy_contour_increment,
canopy_contour_min,
-canopy_contour_increment),
colour = "red", linewidth = 0.25, linetype = "dashed") +
geom_contour(aes(z = value), breaks = 0, colour = "black", linewidth = 0.5) +
scale_x_continuous("Hour", breaks = seq(0,24,6)) +
scale_y_continuous("Canopy cover", breaks = seq(0, 1, 0.25)) +
scale_colour_viridis_c("Selection") +
# ggtitle("Canopy Cover",
# subtitle = "Three pairs of harmonics") +
theme_classic() +
theme(legend.position = "none")
canopy_quad_3p
Combining the plots
Movement parameters
Code
gamma_df <- rbind(gamma_df_0p, gamma_df_1p, gamma_df_2p, gamma_df_3p)
gamma_df <- gamma_df %>% mutate(model_f = as.numeric(factor(model)))
mean_sl <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = mean_sl, group = factor(id)),
alpha = 0.25) +
geom_path(data = gamma_df, aes(x = hour, y = mean, linetype = model)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Mean step length") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle("Observed and modelled mean step length") +
theme_classic() +
theme(legend.position = "bottom")
mean_sl
Code
# ggsave(paste0("outputs/plots/manuscript_figs_R1/mean_sl_",
# Sys.Date(), ".png"),
# width=150, height=90, units="mm", dpi = 1000)
median_sl <- ggplot() +
geom_path(data = movement_summary_buffalo,
aes(x = hour, y = median_sl, group = factor(id)),
alpha = 0.25) +
geom_path(data = gamma_df, aes(x = hour, y = median, linetype = model)) +
scale_x_continuous("Hour", breaks = seq(0,24,2)) +
scale_y_continuous("Median step length") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle("Observed and modelled median step length") +
theme_classic() +
theme(legend.position = "bottom")
median_sl
Habitat selection
Code
harmonics_scaled_long_0p <- harmonics_scaled_long_0p %>% mutate(model = "0p")
harmonics_scaled_long_1p <- harmonics_scaled_long_1p %>% mutate(model = "1p")
harmonics_scaled_long_2p <- harmonics_scaled_long_2p %>% mutate(model = "2p")
harmonics_scaled_long_3p <- harmonics_scaled_long_3p %>% mutate(model = "3p")
harmonics_scaled_long_Mp <- rbind(harmonics_scaled_long_0p,
harmonics_scaled_long_1p,
harmonics_scaled_long_2p,
harmonics_scaled_long_3p)
coef_titles <- unique(harmonics_scaled_long_Mp$coef)
ndvi_harms <- ggplot() +
geom_path(data = harmonics_scaled_long_Mp %>%
filter(coef == "ndvi"),
aes(x = hour, y = value, linetype = model)) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle("NDVI") +
theme_classic() +
theme(legend.position = "bottom")
ndvi_harms
Code
ndvi_2_harms <- ggplot() +
geom_path(data = harmonics_scaled_long_Mp %>%
filter(coef == "ndvi_2"),
aes(x = hour, y = value, linetype = model)) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle(expression(NDVI^2)) +
theme_classic() +
theme(legend.position = "bottom")
ndvi_2_harms
Code
canopy_harms <- ggplot() +
geom_path(data = harmonics_scaled_long_Mp %>%
filter(coef == "canopy"),
aes(x = hour, y = value, linetype = model)) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle("Canopy cover") +
theme_classic() +
theme(legend.position = "bottom")
canopy_harms
Code
canopy_2_harms <- ggplot() +
geom_path(data = harmonics_scaled_long_Mp %>%
filter(coef == "canopy_2"),
aes(x = hour, y = value, linetype = model)) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
scale_y_continuous(expression(beta)) +
scale_x_continuous("Hour") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle(expression(Canopy~cover^2)) +
theme_classic() +
theme(legend.position = "bottom")
canopy_2_harms
Code
herby_harms <- ggplot() +
geom_path(data = harmonics_scaled_long_Mp %>%
filter(coef == "herby"),
aes(x = hour, y = value, linetype = model)) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
scale_y_continuous(expression(beta), limits = c(-0.4,0.15)) +
scale_x_continuous("Hour") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle("Herbaceous vegetation") +
theme_classic() +
theme(legend.position = "bottom")
herby_harms
Code
slope_harms <- ggplot() +
geom_path(data = harmonics_scaled_long_Mp %>%
filter(coef == "slope"),
aes(x = hour, y = value, linetype = model)) +
geom_hline(yintercept = 0, linetype = "dashed", colour = "red") +
scale_y_continuous(expression(beta), limits = c(-0.4,0.15)) +
scale_x_continuous("Hour") +
scale_linetype_manual("Model", breaks=c("0p","1p", "2p", "3p"),
values=c(4,3,2,1)) +
ggtitle("Slope") +
theme_classic() +
theme(legend.position = "bottom")
slope_harms
Code
Combining selection surfaces
NDVI
- A = 0p model
- B = 1p model
- C = 2p model
- D = 3p model
Code
ggarrange(ndvi_quad_0p + theme(plot.title = element_blank(),
axis.title.x = element_blank(),
axis.text.x = element_blank()),
ndvi_quad_1p + theme(plot.title = element_blank(),
axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.title.y = element_blank(),
),
ndvi_quad_2p,
ndvi_quad_3p + theme(plot.title = element_blank(),
axis.title.y = element_blank(),
),
labels = c("A", "B", "C", "D"),
ncol = 2, nrow = 2,
legend = "none",
common.legend = TRUE)
Canopy cover
- A = 0p model
- B = 1p model
- C = 2p model
- D = 3p model
Code
ggarrange(canopy_quad_0p + theme(plot.title = element_blank(),
axis.title.x = element_blank(),
axis.text.x = element_blank()),
canopy_quad_1p + theme(plot.title = element_blank(),
axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.title.y = element_blank(),
),
canopy_quad_2p,
canopy_quad_3p + theme(plot.title = element_blank(),
axis.title.y = element_blank(),
),
labels = c("A", "B", "C", "D"),
ncol = 2, nrow = 2,
legend = "none",
common.legend = TRUE)
Adding all selection surfaces to the same plot
We combine these plots into the plot that is in the paper. On the top is the NDVI selection surface, and on the bottom is the canopy cover selection surface.
Code
surface_plots_0p <- ggarrange(ndvi_quad_0p +
ggtitle("0p") +
theme(axis.title.x = element_blank(),
axis.text.x = element_blank()),
canopy_quad_0p +
scale_x_continuous("Hour", breaks = c(0,12,24)) +
theme(plot.title = element_blank()),
ncol = 1, nrow = 2,
align = "v",
legend = "none",
common.legend = TRUE)Scale for x is already present.
Adding another scale for x, which will replace the existing scale.
Code
surface_plots_1p <- ggarrange(ndvi_quad_1p +
ggtitle("1p") +
theme(axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.title.y = element_blank(),
axis.text.y = element_blank()),
canopy_quad_1p +
theme(plot.title = element_blank(),
axis.title.y = element_blank(),
axis.text.y = element_blank()),
ncol = 1, nrow = 2,
align = "v",
legend = "none",
common.legend = TRUE)
surface_plots_1p
Code
surface_plots_2p <- ggarrange(ndvi_quad_2p +
ggtitle("2p") +
theme(axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.title.y = element_blank(),
axis.text.y = element_blank()),
canopy_quad_2p +
theme(plot.title = element_blank(),
axis.title.y = element_blank(),
axis.text.y = element_blank()),
ncol = 1, nrow = 2,
align = "v",
legend = "none",
common.legend = TRUE)
surface_plots_2p
Code
surface_plots_3p <- ggarrange(ndvi_quad_3p +
ggtitle("3p") +
theme(axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.title.y = element_blank(),
axis.text.y = element_blank()),
canopy_quad_3p +
theme(plot.title = element_blank(),
axis.title.y = element_blank(),
axis.text.y = element_blank()),
ncol = 1, nrow = 2,
align = "v",
legend = "none",
common.legend = TRUE)
surface_plots_3p
All selection surfaces
Code
References
Fieberg, John, Johannes Signer, Brian Smith, and Tal Avgar. 2021. “A ’How to’ Guide for Interpreting Parameters in Habitat-Selection Analyses.” The Journal of Animal Ecology 90 (5): 1027–43. https://doi.org/10.1111/1365-2656.13441.
Forrest, Scott W, Dan Pagendam, Michael Bode, Christopher Drovandi, Jonathan R Potts, Justin Perry, Eric Vanderduys, and Andrew J Hoskins. 2024. “Predicting Fine‐scale Distributions and Emergent Spatiotemporal Patterns from Temporally Dynamic Step Selection Simulations.” Ecography, December. https://doi.org/10.1111/ecog.07421.
Session info
R version 4.4.1 (2024-06-14 ucrt)
Platform: x86_64-w64-mingw32/x64
Running under: Windows 10 x64 (build 19045)
Matrix products: default
locale:
[1] LC_COLLATE=English_Australia.utf8 LC_CTYPE=English_Australia.utf8
[3] LC_MONETARY=English_Australia.utf8 LC_NUMERIC=C
[5] LC_TIME=English_Australia.utf8
time zone: Australia/Brisbane
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] scales_1.3.0 patchwork_1.3.0 MASS_7.3-60.2 ggpubr_0.6.0
[5] beepr_2.0 tictoc_1.2.1 terra_1.7-78 survival_3.6-4
[9] amt_0.2.2.0 lubridate_1.9.3 forcats_1.0.0 stringr_1.5.1
[13] dplyr_1.1.4 purrr_1.0.2 readr_2.1.5 tidyr_1.3.1
[17] tibble_3.2.1 ggplot2_3.5.1 tidyverse_2.0.0
loaded via a namespace (and not attached):
[1] tidyselect_1.2.1 viridisLite_0.4.2 farver_2.1.2 fastmap_1.2.0
[5] digest_0.6.37 timechange_0.3.0 lifecycle_1.0.4 sf_1.0-17
[9] magrittr_2.0.3 compiler_4.4.1 rlang_1.1.4 tools_4.4.1
[13] utf8_1.2.4 yaml_2.3.10 knitr_1.48 ggsignif_0.6.4
[17] labeling_0.4.3 htmlwidgets_1.6.4 bit_4.0.5 classInt_0.4-10
[21] abind_1.4-8 KernSmooth_2.23-24 withr_3.0.1 grid_4.4.1
[25] fansi_1.0.6 e1071_1.7-16 colorspace_2.1-1 isoband_0.2.7
[29] cli_3.6.3 rmarkdown_2.28 crayon_1.5.3 generics_0.1.3
[33] rstudioapi_0.16.0 tzdb_0.4.0 DBI_1.2.3 proxy_0.4-27
[37] audio_0.1-11 splines_4.4.1 parallel_4.4.1 vctrs_0.6.5
[41] Matrix_1.7-0 jsonlite_1.8.8 carData_3.0-5 car_3.1-2
[45] hms_1.1.3 bit64_4.0.5 rstatix_0.7.2 units_0.8-5
[49] glue_1.7.0 codetools_0.2-20 cowplot_1.1.3 stringi_1.8.4
[53] gtable_0.3.5 munsell_0.5.1 pillar_1.9.0 htmltools_0.5.8.1
[57] R6_2.5.1 Rdpack_2.6.1 vroom_1.6.5 evaluate_1.0.0
[61] lattice_0.22-6 rbibutils_2.2.16 backports_1.5.0 broom_1.0.6
[65] class_7.3-22 Rcpp_1.0.13 gridExtra_2.3 xfun_0.47
[69] pkgconfig_2.0.3